A phase-space beam summation formulation for ultrawide-band radiation

A new discrete phase space Gaussian beam (GB) summation representation for ultrawide-band (UWB) radiation from an aperture source distribution is presented. The formulation is based on the theory of the windowed Fourier transform (WFT) frames, wherein we introduce a novel relation between the frequency and the frame overcompleteness. With this procedure, the discrete lattice of beams that are emitted by the aperture satisfies the main requirement of being frequency independent, so that only a single set of beams needs to be traced through the medium for all the frequencies in the band. It is also shown that a properly tuned class of iso-diffracting (ID) Gaussian-windows provides the "snuggest" frame representation for all frequencies, thus generating stable and localized expansion coefficients. Furthermore, due to the ID property, the resulting GBs propagators are fully described by frequency independent matrices whose calculation in the ambient environment need to be done only once for all frequencies. Consequently, the theory may also be expressed directly in the time-domain as will be presented elsewhere. The localization implied by the new formulation is demonstrated numerically for an UWB focused aperture. It is shown that the algorithm extracts the local radiation properties of the aperture source and enhances only those beams that conform with these properties, i.e., those residing near the phase space Lagrange manifold. Further localization is due to the fact the algorithm accounts only for beams that pass within a few beamwidths vicinity of the observation point. It is thus shown that the total number of beams is much smaller than the Landau Pollak bound on the aperture's degrees of freedom.